Exercise 16, § 1

Functions of a Real Variable, II, p. 31

Let $\mathbf{f}$ be a regulated vector function in $I = [a, b] \subset \mathbf{R}$ , $\mathbf{u}$ a primitive of $\mathbf{f}$ in $I$ , $D$ a closed convex set containing $\mathbf{u}(I)$ . Now prove that if $g$ is a monotone numerical function in $I$ , one has

\int_a^b \mathbf{f}(t)g(t) \, dt = (\mathbf{u}(b) - \mathbf{c})g(b) + (\mathbf{c} - \mathbf{u}(a))g(a)

where c belongs to D (reduce to the case where $g$ is a monotone step function). Deduce that if f is a regulated numerical function in $I$ , there exists c ∈ I such that

\int_a^b f(t)g(t)\,dt = g(a) \int_a^c f(t)\,dt + g(b) \int_c^b f(t)\,dt

("second mean theorem").

Solution

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